Core Idea: If the integrand has the form f(g(x))*g'(x), substitute u = g(x) to simplify.

Pattern 1: Linear Substitution integral f(ax+b) dx = $\frac{1}{a}$F(ax+b) + C Example: integral cos(3x+2) dx = $\frac{1}{3}$sin(3x+2) + C

Pattern 2: Function and its Derivative integral [f(x)]^n * f'(x) dx = [f(x)]^$\frac{n+1}{(n+1)}$ + C Example: integral $sin^5$(x)*cos(x) dx = sin^6$$\frac{x}{6} + C (u = sin(x))

Pattern 3: Reciprocal with Derivative integral f'$\frac{x}{f}$(x) dx = ln|f(x)| + C Example: integral $\frac{2x+1}{(x^2+x+3)}$ dx = ln|$x^{2+x+3}$| + C

Pattern 4: Adjusting Constants When the numerator is close to but not exactly the derivative: integral $\frac{4x+7}{(2x^2+7x+3)}$ dx: Note d/dx(2$x^{2+7x+3}$) = 4x+7. Perfect match! = ln|2$x^{2+7x+3}$| + C

When it's not a perfect match, split: integral $\frac{3x+5}{(x^2+4x+1)}$ dx: d/dx($x^{2+4x+1}$) = 2x+4. Write 3x+5 = $\frac{3}{2}$(2x+4) + 5-6 = $\frac{3}{2}$(2x+4) - 1. Split into two integrals.

Pattern 5: Trigonometric Substitutions for Radicals

• sqrt(a^{2-x}^2): x = asin(t), dx = acos(t)dt, sqrt becomes a*cos(t)
• sqrt(a^{2+x}^2): x = atan(t), dx = a$sec^2$(t)dt, sqrt becomes a*sec(t)
• sqrt(x^{2-a}^2): x = asec(t), dx = asec(t)tan(t)dt, sqrt becomes atan(t)

Pattern 6: Reciprocal Substitution For integrals with 1/$x^n$ terms: let x = 1/t, dx = -dt/$t^2$. Useful for integral $\frac{dx}{x^2*sqrt(x^2-1}$) type problems.